In his search for a Supreme Court nominee, President Obama is going into the backyard of Republican Senate Judiciary Chairman Chuck Grassley of Iowa, who has vowed not to hold confirmation hearings on any candidate.

The White House is vetting Jane L. Kelly, 51, a judge on the U.S. Eighth Circuit Court of Appeals in Cedar Rapids, Iowa. The New York Times reported Wednesday that the FBI is conducting background checks on Judge Kelly, citing anonymous sources.

A graduate of the same Harvard Law School class as Mr. Obama in 1991, Judge Kelly was confirmed by the Senate 96-0 in April 2013 for the seat on the Eighth Circuit. At the time, Mr. Grassley praised her and urged colleagues to support her nomination.

But Mr. Grassley said Wednesday his previous support for the judge has no bearing on the current impasse.

“I supported her for the 8th Circuit, but as I’ve said it’s the principle not the person,” Mr. Grassley said in a statement. “If [Hillary] Clinton or [Sen. Bernie] Sanders is elected, she may be on the short list.”

Mr. Grassley added, “Supreme Court scrutiny is some of the toughest there is. Just look at Robert Bork, who was also unanimously confirmed for a circuit court seat.”

Senate Democrats rejected Bork’s nomination to the Supreme Court in 1987; among the lawmakers voting against him was then-Sen. Joseph R. Biden of Delaware, now the vice president.

Mr. Obama is studying potential nominees to replace the late Justice Antonin Scalia, who died last month at age 79. Mr. Grassley and Senate Majority Leader Mitch McConnell, Kentucky Republican, reiterated to the president in a White House meeting on Tuesday that they will not hold confirmation hearings or a vote on any Supreme Court nominee in a presidential election year.

Mr. McConnell said filling the vacancy “is something the American people should decide” by electing a new president in November.

White House press secretary Josh Earnest said as of Wednesday, Mr. Obama and his advisers have reached out to all 100 senators about filling the vacancy. The president is likely to submit a nomination within the next two weeks.

Others who are believed to be under consideration are federal appeals court Judges Srikanth Srinivasan, Patricia Millett and Merrick Garland.

A native of Indiana, Judge Kelly served as a federal public defender in Iowa before becoming a judge. In 2004, she was attacked while jogging in Cedar Rapids and left for dead. Her assailant was never captured.